What is Orthogonal Projection?

Orthogonal projection is a fundamental concept in linear algebra and vector mathematics. It refers to the process of projecting one vector onto another vector or onto a subspace. In simpler terms, it’s a way to find the component of one vector that points in the same or opposite direction as another vector.

When we project vector u onto vector v, we’re essentially finding the “shadow” that u would cast if light were shining perpendicular to v. This projection is crucial in many areas of mathematics, physics, and engineering, including computer graphics, signal processing, and data analysis.

The orthogonal projection of u onto v is given by the formula:

proj_v(u) = (u · v / ||v||^2) * v

Where u · v is the dot product of u and v, and ||v|| is the magnitude (length) of vector v.

How to Use the Orthogonal Projection Calculator

Step 1: Enter the components of vector u

Begin by inputting the three components of vector u into the designated fields labeled u1, u2, and u3. These represent the x, y, and z components of the vector in three-dimensional space. For example, if your vector u is (2, 3, -1), enter 2 for u1, 3 for u2, and -1 for u3.

Step 2: Input the components of vector v

Next, fill in the components of vector v in the fields labeled v1, v2, and v3. This is the vector onto which you want to project vector u. For instance, if vector v is (1, -2, 4), enter 1 for v1, -2 for v2, and 4 for v3.

Step 3: Click the “Calculate Projection” button

After entering all the components for both vectors, click the blue “Calculate Projection” button. The calculator will process your input and compute the orthogonal projection.

Step 4: Interpret the results

The calculator will display the result of the orthogonal projection as a new vector. This vector represents the component of u that is parallel to v. The result will be shown in the format (x, y, z), where x, y, and z are the components of the projected vector.

Step 5: Review the calculation method

Below the result, you’ll find an explanation of how the projection was calculated. This includes the intermediate steps such as the dot product of u and v, the magnitude squared of v, and the scalar factor used in the projection. This information can help you understand the process and verify the calculations if needed.

By following these steps, you can easily compute orthogonal projections for any pair of three-dimensional vectors, making complex vector calculations simple and accessible.